A Quasi-Newton Method with Rank-Two Update to Solve Interval Optimization Problems

Debdas Ghosh1
1Department of Mathematics, Indian Institute of Information Technology Kalyani, Kalyani, India

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Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 3rd edn. Wiley, New York (2006)

Bhurjee, A.K., Panda, G.: Efficient solution of interval optimization problem. Math. Methods Oper. Res. 76(3), 273–288 (2012)

Boggs, P.T., Byrd, R.H., Schnabel, R.B.: Numerical optimization 1984. In: Proceedings of the SIAM Conference on Numerical Optimization. Boulder, Colorado, June 12–14, 1984, Vol. 20, Siam (1985)

Chakraborty, D., Ghosh, D.: Analytical fuzzy plane geometry II. Fuzzy Sets Syst. 243, 84–100 (2014)

Chalco-Cano, Y., Rufian-Lizana, A., Roman-Flores, H., Jimenez-Gamero, M.D.: Calculus for interval-valued functions using generalized Hukuhara derivative and applications. Fuzzy Sets Syst. 219, 49–67 (2013)

Chalco-Cano, Y., Silva, G.N., Rufian-Lizana, A.: On the Newton method for solving fuzzy optimization problems. Fuzzy Sets Syst. 272, 60–69 (2015)

Ghosh, D.: A Newton method for capturing efficient solutions of interval optimization problems. Opsearch (2016). doi: 10.1007/s12597-016-0249-6

Ghosh, D.: Newton method to obtain efficient solutions of the optimization problems with interval-valued objective functions. J. Appl. Math. Comput. (2016). doi: 10.1007/s12190-016-0990-2

Ghosh, D., Chakraborty, D.: A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. Fuzzy Sets Syst. 272, 1–29 (2015)

Ghosh, D., Chakraborty, D.: A method for capturing the entire fuzzy non-dominated set of a fuzzy multi-criteria optimization problem. J. Intel. Fuzzy Syst. 26, 1223–1234 (2014)

Ghosh, D., Chakraborty, D.: Quadratic interpolation technique to minimize univariable fuzzy functions. Int. J. Appl. Comput. Math. (2015). doi: 10.1007/s40819-015-0123-x

Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry I. Fuzzy Sets Syst. 209, 66–83 (2012)

Ghosh, D., Chakraborty, D.: Analytical fuzzy plane geometry III. Fuzzy Sets Syst. 283, 83–107 (2016)

Ghosh, D., Chakraborty, D.: On general form of fuzzy lines and its application in fuzzy line fitting. J. Intel. Fuzzy Syst. 29, 659–671 (2015)

Ghosh, D., Chakraborty, D.: A direction based classical method to obtain complete Pareto set of multi-criteria optimization problems. Opsearch 52(2), 340–366 (2015)

Ghosh, D., Chakraborty, D.: A new Pareto set generating method for multi-criteria optimization problems. Oper. Res. Lett. 42, 514–521 (2014)

Ghosh, D., Chakraborty, D.: Ideal Cone: a new method to generate complete pareto set of multi-criteria optimization problems. In: Mathematics and Computing 2013, Vol. 91, Springer, Proceedings in Mathematics and Statistics, pp. 171–190 (2013)

Hansen, W.G.E.: Global Optimization Using Interval Analysis. Marcel Dekker Inc., New York (2004)

Higham, N.J.: Accuracy and Stability of Numerical Algorithms. SIAM (2002)

Hladik, M.: Interval Linear Programming: A Survey. Nova Science Publishers, New York (2012)

Hu, B., Wang, S.: A novel approach in uncertain programming, part I: new arithmetic and order relation of interval numbers. J. Ind. Manag. Optim. 2(4), 351–371 (2006)

Hu, B., Wang, S.: A novel approach in uncertain programming, part II: a class of constrained nonlinear programming with interval objective function. J. Ind. Manag. Optim. 2(4), 373–385 (2006)

Hukuhara, M.: Integration des applications mesurables dont la valeur est un compact convexe. Funkc Ekvacioj 10, 205–223 (1967)

Ishibuchi, H., Tanaka, H.: Multiobjective programming in optimization of the interval objective function. Eur. J. Oper. Res. 48(2), 219–225 (1990)

Jayswal, A., Stancu-Minasian, I., Ahmed, I.: On sufficiency and duality for a class of interval-valued programming problems. Appl. Math. Comput. 218(8), 4119–4127 (2011)

Jeyakumar, V., Li, G.Y.: Robust duality for fractional programming problems with constraint-wise data uncertainty. Eur. J. Oper. Res. 151(2), 292–303 (2011)

Jiang, C., Han, X., Liu, G.R.: A nonlinear interval number programming method for uncertain optimization problems. Eur. J. Oper. Res. 188(1), 1–13 (2008)

Li, W., Tian, X.: Numerical solution method for general interval quadratic programming. Appl. Math. Comput. 202(2), 589–595 (2008)

Liu, S.T., Wang, R.T.: A numerical solution method to interval quadratic programming. Appl. Math. Comput. 189(2), 1274–1281 (2007)

Luciana, T.G., Barrosb, L.C.: A note on the generalized difference and the generalized differentiability. Fuzzy Sets Syst. 280, 142–145 (2015)

Marin, M.: On existence and uniqueness in thermoelasticity of micropolar bodies. C. R. Acad. Sci. Paris Ser. II 321(12), 475–480 (1995)

Marin, M.: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 40(3), 1391–1399 (1999)

Marin, M., Agarwal, R.P., Mahmoud, S.R.: Nonsimple material problems addressed by the Lagrange’s identity. Bound. Value Prob. 1–14, Article No. 135 (2013)

Markov, S.: Calculus for interval functions of a real variable. Computing 22, 325–337 (1979)

Moore, R.: Interval Anal. Prentice-Hall, Englewood Cliffs (1966)

Moore, R.E., Kearfott, R.B., Cloud, M.J.: Introduction to Interval Analysis. SIAM (2009)

Neumaier, A.: Interval Methods for Systems of Equation, Encyclopedia of Mathematics and Its Applications, vol. 37. Cambridge University Press, Cambridge (1990)

Nocedal, J.: Updating quasi-Newton matrices with limited storage. Math. Comput. 35, 773–782 (1980)

Nocedal, J., Stephen, W.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)

Pirzada, U.M., Pathak, V.D.: Newton Method for solving the multi-variable fuzzy optimization problem. J. Optim. Theory Appl. 156, 867–881 (2013)

Rohn, J.: Positive definiteness and stability of interval matrices. SIAM J. Matrix Anal. Appl. 15, 175–184 (1994)

Sengupta, A., Pal, T.K.: Fuzzy Preference Ordering of Interval Numbers in Decision Problems. Series on Studies in Fuzziness and Soft Computing, vol. 238. Springer, Berlin (2009)

Stahl, T.: Interval methods for bounding the range of Polynomials and solving sytems of nonlinear equation. PhD thesis. Johannes Kepler University Linz, Austria (1994)

Stefanini, L.: A Generalization of Hukuhara Difference, Soft Methods for Handling Variability and Imprecision. Series on Advances in Soft Computing, vol. 48, pp. 203–210. Springer, Berlin (2008)

Stefanini, L.: A generalization of Hukuhara difference and division for interval and fuzzy arithmetic. Fuzzy Sets Syst. 161, 1564–1584 (2010)

Wang, H., Zhang, R.: Optimality conditions and duality for arcwise connected interval optimization problems. Opsearch 52(4), 870–883 (2015)

Wu, H.C.: On interval-valued nonlinear programing problem. J. Math. Anal. Appl. 338(1), 299–316 (2008)